Tate’s non-archimedean uniformization of elliptic curves
E896827
Tate’s non-archimedean uniformization of elliptic curves is a foundational theory in arithmetic geometry that describes certain elliptic curves over non-archimedean fields via analytic uniformization using formal q-expansions, leading to what are now called Tate curves.
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theory
ⓘ
theory in arithmetic geometry ⓘ uniformization theory ⓘ |
| appliesTo |
elliptic curves over non-archimedean fields
ⓘ
elliptic curves with split multiplicative reduction ⓘ |
| characterizes |
elliptic curves with non-integral j-invariant
ⓘ
elliptic curves with split multiplicative reduction over local fields ⓘ |
| context |
elliptic curves over complete non-archimedean fields
ⓘ
elliptic curves over p-adic fields ⓘ |
| describes |
elliptic curves as quotients of the multiplicative group
ⓘ
elliptic curves via analytic uniformization ⓘ |
| developedBy | John Tate NERFINISHED ⓘ |
| field |
arithmetic geometry
ⓘ
non-archimedean analytic geometry ⓘ p-adic analysis ⓘ |
| foundationFor |
non-archimedean analytic uniformization methods
ⓘ
theory of Tate curves NERFINISHED ⓘ |
| generalizedBy |
Mumford curves theory
ⓘ
Raynaud’s p-adic uniformization of abelian varieties ⓘ |
| hasOutcome |
classification of certain elliptic curves via q-parameters
ⓘ
explicit formulas for invariants of elliptic curves in terms of q ⓘ |
| influenced |
development of rigid analytic geometry
ⓘ
p-adic uniformization of abelian varieties ⓘ |
| involves |
parameter q with |q|<1 in a non-archimedean field
ⓘ
quotient of the multiplicative group by a discrete subgroup ⓘ |
| namedAfter | John Tate NERFINISHED ⓘ |
| produces | Tate curves NERFINISHED ⓘ |
| provides |
analytic parametrization of points on elliptic curves
ⓘ
explicit q-parameter for elliptic curves ⓘ |
| relatesTo |
Galois representations attached to elliptic curves
ⓘ
Néron models of elliptic curves ⓘ Weierstrass equation of elliptic curves NERFINISHED ⓘ j-invariant of elliptic curves ⓘ |
| timePeriod | 1960s ⓘ |
| usedIn |
Iwasawa theory of elliptic curves
ⓘ
computation of conductors of elliptic curves ⓘ p-adic modular forms ⓘ study of local L-factors of elliptic curves ⓘ theory of q-expansions of modular forms ⓘ |
| usesConcept |
formal power series
ⓘ
non-archimedean absolute value ⓘ p-adic analytic functions ⓘ q-expansion ⓘ rigid analytic geometry ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.